Theorems of Moment of Inertia

There are two theorems which connect moments of inertia about two axes, one of which is passing through the Centre of Mass of the body:

  1. The theorem of parallel axes
  2. The theorem of perpendicular axes

Theorem of Parallel Axes

Suppose the given rigid body rotates about an axis passing through any point P other than the centre of mass. The moment of inertia about this axis can be found from a knowledge of the moment of inertia about a parallel axis through the centre of mass.

Theorem of parallel axis states that the moment of inertia about an axis parallel to the axis passing through its centre of mass is equal to the moment of inertia about its centre of mass plus the product of mass and square of the perpendicular distance between the parallel axes. If I denotes the required moment of inertia and IC denotes the moment of inertia about a parallel axis passing through the Centre of Mass, then

I = IC + Md2

where M is the mass of the body and d is the distance between the two axes. 

Theorem of Perpendicular Axes

Three mutually perpendicular axes, two of which, say x and y are in the plane of the body, and the third, the z axis, is perpendicular to the plane. The perpendicular axes theorem states that the sum of the moments of inertia about x and y axes is equal to the moment of inertia about the z axis.

Iz = Ix + Iy